Length
   HOME

TheInfoList



OR:

Length is a measure of
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
. In the
International System of Quantities The International System of Quantities (ISQ) is a standard system of Quantity, quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This ...
, length is a quantity with
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
distance. In most
systems of measurement A system of units of measurement, also known as a system of units or system of measurement, is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defi ...
a base unit for length is chosen, from which all other units are derived. In the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI) system, the base unit for length is the
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
. Length is commonly understood to mean the most extended
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include
height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For an example of vertical extent, "This basketball player is 7 foot 1 inches in height." For an e ...
, which is vertical length or vertical extent, width, breadth, and depth. ''Height'' is used when there is a base from which vertical measurements can be taken. ''Width'' and ''breadth'' usually refer to a shorter dimension than ''length''. ''Depth'' is used for the measure of a third dimension. Length is the measure of one spatial dimension, whereas
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
is a measure of two dimensions (length squared) and
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
is a measure of three dimensions (length cubed).


History

Measurement has been important ever since humans settled from nomadic lifestyles and started using building materials, occupying land and trading with neighbours. As trade between different places increased, the need for standard units of length increased. And later, as society has become more technologically oriented, much higher accuracy of measurement is required in an increasingly diverse set of fields, from micro-electronics to interplanetary ranging. Under Einstein's
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
, length can no longer be thought of as being constant in all reference frames. Thus a
ruler A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instr ...
that is one metre long in one frame of reference will not be one metre long in a reference frame that is moving relative to the first frame. This means the length of an object varies depending on the speed of the observer.


Use in mathematics


Euclidean geometry

In Euclidean geometry, length is measured along straight lines unless otherwise specified and refers to segments on them. Pythagoras's theorem relating the length of the sides of a right triangle is one of many applications in Euclidean geometry. Length may also be measured along other types of curves and is referred to as arclength. In a
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
, the length of an
altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...
, a line segment drawn from a vertex
perpendicular In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
to the side not passing through the vertex (referred to as a base of the triangle), is called the height of the triangle. The
area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
of a
rectangle In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
is defined to be length × width of the rectangle. If a long thin rectangle is stood up on its short side then its area could also be described as its height × width. The
volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...
of a solid rectangular box (such as a plank of wood) is often described as length × height × depth. The
perimeter A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimet ...
of a
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
is the sum of the lengths of its sides. The
circumference In geometry, the circumference () is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length arou ...
of a circular disk is the length of the boundary (a
circle A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
) of that disk.


Other geometries

In other geometries, length may be measured along possibly curved paths, called geodesics. The
Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...
used in
general relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
is an example of such a geometry. In
spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applicati ...
, length is measured along the great circles on the sphere and the distance between two points on the sphere is the shorter of the two lengths on the great circle, which is determined by the plane through the two points and the center of the sphere.


Graph theory

In an unweighted graph, the length of a cycle, path, or
walk Walking (also known as ambulation) is one of the main gaits of terrestrial locomotion among legged animals. Walking is typically slower than running and other gaits. Walking is defined as an "inverted pendulum" gait in which the body vaults over ...
is the number of edges it uses. In a weighted graph, it may instead be the sum of the weights of the edges that it uses. Length is used to define the shortest path, girth (shortest cycle length), and longest path between two vertices in a graph.


Measure theory

In measure theory, length is most often generalized to general sets of \mathbb^n via the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. In the one-dimensional case, the Lebesgue outer measure of a set is defined in terms of the lengths of open intervals. Concretely, the length of an
open interval In mathematics, a real interval is the set (mathematics), set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without ...
is first defined as : \ell(\)=b-a. so that the Lebesgue outer measure \mu^*(E) of a general set E may then be defined as : \mu^*(E)=\inf\left\.


Units

In the physical sciences and engineering, when one speaks of , the word is synonymous with
distance Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
. There are several units that are used to measure length. Historically, units of length may have been derived from the lengths of human body parts, the distance travelled in a number of paces, the distance between landmarks or places on the Earth, or arbitrarily on the length of some common object. In the
International System of Units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official s ...
(SI), the base unit of length is the
metre The metre (or meter in US spelling; symbol: m) is the base unit of length in the International System of Units (SI). Since 2019, the metre has been defined as the length of the path travelled by light in vacuum during a time interval of of ...
(symbol, m), now defined in terms of the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
(about 300 million metres per
second The second (symbol: s) is a unit of time derived from the division of the day first into 24 hours, then to 60 minutes, and finally to 60 seconds each (24 × 60 × 60 = 86400). The current and formal definition in the International System of U ...
). The
millimetre 330px, Different lengths as in respect of the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is between 1 metre to 1 millimetre. The millimetre (American and British English spelling differences#-re, -er, i ...
(mm),
centimetre upright=1.35, Different lengths as in respect to the electromagnetic spectrum, measured by the metre and its derived scales. The microwave is in-between 1 meter to 1 millimeter. A centimetre (International spelling) or centimeter (American ...
(cm) and the
kilometre The kilometre (SI symbol: km; or ), spelt kilometer in American English, American and Philippine English, is a unit of length in the International System of Units (SI), equal to one thousand metres (kilo- being the SI prefix for ). It is the ...
(km), derived from the metre, are also commonly used units. In U.S. customary units, English or imperial system of units, commonly used units of length are the
inch The inch (symbol: in or prime (symbol), ) is a Units of measurement, unit of length in the imperial units, British Imperial and the United States customary units, United States customary System of measurement, systems of measurement. It is eq ...
(in), the
foot The foot (: feet) is an anatomical structure found in many vertebrates. It is the terminal portion of a limb which bears weight and allows locomotion. In many animals with feet, the foot is an organ at the terminal part of the leg made up o ...
(ft), the
yard The yard (symbol: yd) is an English units, English unit of length in both the British imperial units, imperial and US United States customary units, customary systems of measurement equalling 3 foot (unit), feet or 36 inches. Sinc ...
(yd), and the
mile The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a imperial unit, British imperial unit and United States customary unit of length; both are based on the older English unit of Unit of length, le ...
(mi). A unit of length used in
navigation Navigation is a field of study that focuses on the process of monitoring and controlling the motion, movement of a craft or vehicle from one place to another.Bowditch, 2003:799. The field of navigation includes four general categories: land navig ...
is the
nautical mile A nautical mile is a unit of length used in air, marine, and space navigation, and for the definition of territorial waters. Historically, it was defined as the meridian arc length corresponding to one minute ( of a degree) of latitude at t ...
(nmi). km = miles Units used to denote distances in the vastness of space, as in
astronomy Astronomy is a natural science that studies celestial objects and the phenomena that occur in the cosmos. It uses mathematics, physics, and chemistry in order to explain their origin and their overall evolution. Objects of interest includ ...
, are much longer than those typically used on Earth (metre or kilometre) and include the
astronomical unit The astronomical unit (symbol: au or AU) is a unit of length defined to be exactly equal to . Historically, the astronomical unit was conceived as the average Earth-Sun distance (the average of Earth's aphelion and perihelion), before its m ...
(au), the
light-year A light-year, alternatively spelled light year (ly or lyr), is a unit of length used to express astronomical distances and is equal to exactly , which is approximately 9.46 trillion km or 5.88 trillion mi. As defined by the International Astr ...
, and the
parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (AU), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...
(pc). Units used to denote sub-atomic distances, as in
nuclear physics Nuclear physics is the field of physics that studies atomic nuclei and their constituents and interactions, in addition to the study of other forms of nuclear matter. Nuclear physics should not be confused with atomic physics, which studies th ...
, are much smaller than the millimetre. Examples include the fermi (fm).


See also

*
Arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
* Humorous units of length * Length measurement *
Metric system The metric system is a system of measurement that standardization, standardizes a set of base units and a nomenclature for describing relatively large and small quantities via decimal-based multiplicative unit prefixes. Though the rules gover ...
* Metric units *
Orders of magnitude (length) The following are examples of order of magnitude, orders of magnitude for different lengths. Overview Detailed list To help compare different orders of magnitude, the following list describes various lengths between 1.6 \times 10^ me ...
*
Reciprocal length Reciprocal length or inverse length is a quantity or measurement used in several branches of science and mathematics, defined as the reciprocal of length. Common units used for this measurement include the reciprocal metre or inverse metre (symbo ...


References

{{Authority control Physical quantities SI base quantities